Finite element analysis (FEA) is a computerized method widely used in industry to model and solve engineering problems relating to complex systems such as three-dimensional non-linear structural design and analysis. FEA derives its name from the manner in which the geometry of the object under consideration is specified. With the advent of the modern digital computer, FEA has been implemented as FEA software. Basically, the FEA software is provided with a model of the geometric description and the associated material properties at each point within the model. In this model, the geometry of the system under analysis is represented by solids, shells and beams of various sizes, which are called elements. The vertices of the elements are referred to as nodes. The model is comprised of a finite number of elements, which are assigned a material name to associate the elements with the material properties. The model thus represents the physical space occupied by the object under analysis along with its immediate surroundings. The FEA software then refers to a table in which the properties (e.g., stress-strain constitutive equation, Young's modulus, Poisson's ratio, thermo-conductivity) of each material type are tabulated. Additionally, the conditions at the boundary of the object (i.e., loadings, physical constraints, etc.) are specified. In this fashion a model of the object and its environment is created.
FEA has two solution techniques: the implicit finite element analysis (“the implicit method”) and the explicit finite element analysis (“the explicit method”). Both methods are used to solve transient dynamic equations of motion and thus obtain an equilibrium solution to the equations. The methods march from time (t) through a discrete time interval or time interval Δt, to time (t+Δt). Such methods are sometimes referred to as time-marching simulation, which contains a number of consecutive time steps or solution cycles.
The present invention relates to the explicit method, which is stable only if time step size is very small—specifically, the time interval must be smaller than the time taken for an elastic wave to propagate from one side of an element to the other. The maximum time step for maintaining a stable solution in the explicit method is referred to as the critical time step size Δtcr. The speed of the elastic wave is a function of material mass and stiffness of the structure represented by the finite element and the element size or dimension. For a FEA model having substantially similar material, the smallest element generally controls the critical time step size.
Even one substantially smaller element in a FEA model can cause the critical time step size unnecessarily small for majority of the elements in the FEA model. This could happen when any element deforms to become too small in the middle of a time-marching simulation. As a result, a very small time step would be required for the remaining of the simulation. Not only would the simulation become very real time consuming, but also impractical. For example, it takes too long to obtain a simulated result for engineer to make design decision in time to meet the commercial need.
A number of prior art approaches to this problem have not been successful, for example, one approach referred to as subcycling that permits different time step sizes to be used in different parts of the finite element model. However, subcycling has its shortcomings and problems, for example, it requires periodically sorting of all elements of the entire FEA model into a number of groups being processed with dynamically changing time step sizes. This is not only difficult to implement in software, but also very costly. Periodical sorting of finite elements requires a significant amount of computer resource often defeating the originally purpose of speeding up the simulation.
Another approach is referred to as mass scaling, which increases the critical time step size of an element by artificially increasing its mass density. However, arbitrarily applying mass scaling to the FEA model can change the dynamic behaviors of the structure (i.e., artificially higher mass) in an unwanted manner.
It would therefore be desirable to have methods and systems for numerically simulating structural behaviors of a product using an explicit FEA with a combined technique of subcycling and mass scaling, such that the simulation is performed efficiently and effectively.